The discrete tuberculosis transmission model with treatment of latently infected individuals
- Hui Cao^{1}Email author and
- Hongwu Tan^{1}
https://doi.org/10.1186/s13662-015-0505-8
© Cao and Tan 2015
Received: 1 December 2014
Accepted: 17 May 2015
Published: 3 June 2015
Abstract
A discrete tuberculosis model with direct progression and treatment of latently infected individuals is presented. The model does not consider the drug-resistant TB, and it assumes that latently infected individuals develop the active disease only because of being endogenous reactive, and a small fraction of infected individuals is assumed to develop the active disease soon after infection. The global stability of a disease-free equilibrium, the persistence of system, and the local stability of endemic equilibrium are discussed. The basic reproductive numbers with different control measures are determined and analyzed, and we give the critical value of probability of successful detection and treatment of infectious individuals. If a treatment only of infectious individuals cannot control TB transmission, the treatment of latent TB individuals should be carried out, and we give the critical value of the probability of treatment of infectious individuals. Numerical simulations are done to demonstrate the complex dynamics of the model.
Keywords
1 Introduction
Differential equations and difference equations are widely applied in epidemiological modeling. They are two typical mathematical approaches for modeling infectious diseases. Since the theory and method for dynamical studies of differential equations have developed much more completely than those for difference equations, there are relatively few difference equations in epidemiological modeling compared with differential equations. In recent years, there were increasing interest and research results on discrete epidemic models [1–8]. The fact that the epidemiological data are usually collected in discrete time units, such as days, weeks or months, makes the discrete model a natural choice to describe a disease transmission. The straightforward recurrence relationship of the difference equation models is easier to understand, which is also a prominent advantage over the differential equation models. The direct comparison of the model results with the actual data provides us a fast and simple way to validate the model structure and parameter estimation. The fact that the discrete models exhibit a richer dynamical behavior than the continuous models brings about more challenging problems for researches, and more interesting results can be obtained. For example, the simple logistic model, \(x_{n+1}=rx_{n}(1-x_{n}/K)\), the Ricker model \(x_{n+1}=x_{n}e^{r-x_{n}/K}\) [9–11], and the Hassell model, \(x_{n+1}=\lambda x_{n}(1+a x_{n})^{-b}\) [12] exhibit a rich dynamical behavior.
There have been increasing interest and more studies on the discrete time epidemic models recently. Various discrete epidemic models have been successfully applied to describe the infectious disease transmission, such as SARS, tuberculosis, HIV/AIDS [13–18]. The theoretical study of discrete epidemic models focuses on the computation of the basic reproduction number [19–21], the existence and the global stability of the disease-free equilibrium [4, 5, 14–16], the existence and the local stability of the endemic equilibrium [22, 23], and the persistence of the disease [14, 15]. Attention has also be paid to various bifurcations of the discrete epidemic models, the equilibrium bifurcation [3, 21, 24–26], the transcritical bifurcation, the flip bifurcation, the saddle-node bifurcation, the Hopf bifurcation, and the bifurcation to chaos [3–5].
There also have been many researches studying mathematical models of the transmission dynamics of TB in human populations. Those researches include slow and fast progression, a variable latent period, MDR TB, multiple strains, exogenous reinfection, generalized households, co-infection with HIV, and the control strategy for TB [27–35]. However, there are a few papers using discrete mathematical models to study the treatment of latent TB as a TB control strategy [3, 36, 37]. The treatment of latent tuberculosis infection is essential to controlling and eliminating TB, because it substantially reduces the risk from latent TB to active TB cases. In this paper, we use the discrete models with direct progression and treatment of latently infected individuals to analyze the impact of the treatment of latently infected individuals on TB transmission.
Our discrete model with treatment of latent TB individuals is presented in the next section. The positivity of solutions is also discussed. The global stability of the disease-free equilibrium and the persistence of the system is discussed in Section 3. The stability of endemic equilibrium is proved in Section 4. The effect of two treatment strategies is studied in Section 5. Numerical simulations are done on the basis of the TB infection data in China to show the effect of treatment strategies. The last section includes concluding remarks and discussions.
2 The discrete TB model
TB is an airborne infectious disease transmitting from person to person via droplets with TB bacilli. After being infected, most people become latently infectious, with the bacteria being alive in the body but inactive. Latently infectious individuals do not have symptoms and cannot spread the infection to others. Most people are able to fight the infection with their immune system, but those people with latent infection are at risk of developing the active disease. Based on these characteristics, there have been many continuous models to be used to describe TB transmission. Here, we will build the discrete mathematical model for TB transmission which considers the direct progression, chemoprophylaxis for the latent individuals, and treatment of the infectious individuals.
The total population is epidemiologically divided into susceptible, latent, and infectious classes. Let \(S(t)\) be the number of susceptible individuals at time t, \(L(t)\) be the number of latent individuals, and \(I(t)\) be the number of infectious individuals. \(N(t)=S(t)+L(t)+I(t)\) is the total population size. We assume that individuals recovered by successful treatment do not acquire immunity, and they will become member of the susceptible compartment. Hence our model is of SLIS type.
In the following, we first discuss the positivity and boundedness of solutions for system (1). The following theorem holds.
Theorem 2.1
The solutions \(S(t)\), \(L(t)\), and \(I(t)\) of system (1) with initial value \(S(0)=S_{0}\geq0\), \(L(0)=L_{0}\geq0\), and \(I(0)=I_{0}\geq0\), respectively, are non-negative for all \(t\geq0\), \(t\in N\). For the system (1), the region Ω is positively invariant and all solutions starting in the Ω approach, enter, or stay in Ω.
Proof
The basic reproductive number \(R_{0}\) is defined mathematically as the spectral radius of the next generation matrix in [39]; in fact, each term in \(R_{0}\) has a clear epidemiological interpretation. \(1/ (1-p(1-\gamma) )\) is the average infection period. \((1-m)p\alpha/ (1-p(1-km)+p\alpha(1-m) )\) is the proportion of latent individuals that become infectious by natural progression. \(p\beta/ (1-p(1-\gamma) )\) is the average of new cases generated by a typical infectious member in the entire infection period, where \(qp\beta/ (1-p(1-\gamma) )\) is the average of new cases generated by a typical infectious member who enters the infectious compartment by natural progression in the entire infection period, \((1-q)p\beta/ (1-p(1-\gamma) )\) is the average of new cases generated by a typical infectious member who enters the infectious compartment by direct progression in the entire infection period.
3 The extinction and persistence for the disease
3.1 The extinction for the disease
Theorem 3.1
If \(R_{0}<1\), then the disease-free equilibrium \(P_{0}^{*}\) of model (3) is globally asymptotically stable; if \(R_{0}>1\), then \(P_{0}^{*}\) is unstable.
Proof
3.2 The persistence for the disease
Theorem 3.2
Proof
We denote \(\mathcal{X}=\Omega\), \(\mathcal{X}_{0}=\{(L,I)\in\mathcal {X}\mid L>0, I>0\}\), and \(\partial\mathcal{X}_{0}=\mathcal{X}\backslash \mathcal{X}_{0}\). Let \(\Phi: \mathcal{X}\rightarrow\mathcal{X}\), \(\Phi_{t}(x_{0})=\phi (t,x_{0})\) be the solution map of model (3) with \(\phi(0, x_{0})=x_{0}\), and \(x_{0}= (L(0), I(0) )\).
It is clear that \(M_{\partial}=\{(0,0)\}\). Furthermore, there is exactly one fixed point \(P_{0}^{*}=(0,0)\) in \(M_{\partial}\). Because \(N^{*}\) is globally attractive in \(\partial X_{0}\) and due to Lemma 5.9 in [41], we know that no subset of \(\mathcal{M}\) forms a cycle in \(\partial \mathcal{X}_{0}\). The definition of \(M_{\partial}\) implies that \(M_{\partial}\) is the maximum positive invariant set in \(\partial\mathcal{X}_{0}\), that is, \(\Phi_{t}(M_{\partial})\subset M_{\partial}\). Therefore, \(\bigcup_{(L(0),I(0))\in M_{\partial}}=P_{0}^{*}\).
\(R_{0}>1\) implies that \(p\beta+p(1-\gamma)>1\), namely, \(p\beta+p(1-\gamma )-\frac{p\beta\epsilon}{N^{*}}>1\) holds for a small \(\epsilon>0\). Therefore, \(H_{1}(t)\rightarrow\infty\) as \(t\rightarrow\infty\) and \(\epsilon\rightarrow0\). The comparison principle implies that \(H(t)\rightarrow\infty\) as \(t\rightarrow\infty\) and \(\epsilon\rightarrow0\). In fact, \(H(t)\leq2\epsilon\) for all \(t\geq t_{1}\), \(t\in N\), a contradiction. It implies that \(L(t)+I(t)\rightarrow\infty\), for all \(t\geq t_{1}\), \(t\in N\). On the other hand, \(0\leq L(t)\leq\epsilon\), \(0\leq I(t)\leq\epsilon\) for all \(t\geq t_{1}\), \(t\in N\). It implies that \(L(t)\rightarrow\infty\) and \(I(t)\rightarrow\infty\) at least established for all \(t\geq t_{1}\), \(t\in N\). There is a contradiction, that is, the conclusion in (4) holds.
4 The stability of the endemic equilibrium
Theorem 4.1
If \(R_{0}>1\), then the endemic equilibrium \(P_{1}^{*}\) of model (3) is locally asymptotically stable.
Proof
5 Effect of control strategies
The two equations of (5) illustrate that the basic reproductive number \(R_{0}\) decreases as the proportion of m and the probability k of successful detection and effective treatment of latently infected individuals. The treatment of latent tuberculosis infection contributes to slowing down TB transmission because it substantially reduces the risk that TB infection will progress to TB disease. Therefore, we consider both treatment of the latently infected individuals and for the infectious individuals. We assume that the word ‘control’ would mean bringing down the number of infectious TB cases.
5.1 Effect of treatment of the infectious individuals
Because the latently TB infected individuals cannot infect others, one does not take to treatment of latently infected individuals in developing and undeveloped countries so as to save expense. Therefore, we only consider treatment of the TB infectious individuals. We will give the critical value \(\gamma^{*}\) of the probability γ of successful detection and treatment of infectious individuals so that the basic reproductive number of the model is less than one.
5.2 Effect of treatment of the latent individuals
When the probability γ of successful detection and treatment of infectious individuals does not reach the critical value \(\gamma^{*}\), we also take the treatment of latently TB individuals, and give the critical value \(m^{*}\) of the probability m of the treatment of latently individuals, so that \(R_{0}<1\).
Because of \(R_{2}>1\), we obtain \(p\alpha qp\beta> (1-p(1-\alpha) ) (1-p(1-\gamma)-(1-q)p\beta )\). It is clear that \((1-p(1-\alpha) ) (1-p(1-\gamma)-(1-q)p\beta )> (p(1-k)-p(1-\alpha) ) ((1-q)p\beta-(1-p(1-\gamma)) )\), which implies \(m^{*}>0\). By (11), we know that if the effective treatment of the individuals in the infectious compartment satisfies \(0<\gamma<\gamma^{*}\), we should strengthen control measures so as to have \(R_{0}<1\). Therefore, we also take the treatment of latent TB individuals, and the probability for receiving the treatment m satisfies \(m>m^{*}\).
6 Conclusion and discussion
In this paper, we analyze a class of discrete SLIS models with direct progression and chemoprophylaxis for latent TB individuals. The basic reproductive numbers \(R_{0}\) with both treatment of latently infected individuals and infectious individuals are determined. Furthermore, we study the global stability of the disease-free equilibrium as \(R_{0}<1\), the persistence of the system, and the local stability of the endemic equilibrium as \(R_{0}>1\). The numerical simulation shows that when \(\beta>1\), and β is not large enough, the solutions of the model are still positive, and the endemic equilibrium may lose the stability and there exists chaos as β increases, which can lead to \(R_{0}\) increasing. By analyzing \(R_{0}\), we learn that the treatment of latent TB individuals contributes to a decrease of the basic reproductive number, namely, a reduction of the speed of TB transmission.
In addition, the basic reproductive number \(R_{1}\) without treatment, and the basic reproductive number \(R_{2}\) with treatment of infectious individuals are determined, respectively. We also discuss the effect of the treatment of the infectious individuals on TB transmission, and give the critical value \(\gamma^{*}\) of the probability γ of successful detection and treatment of infectious individuals so that the basic reproductive number of the model is \(R_{2}<1\). Once \(R_{2}>1\), that is, \(\gamma<\gamma^{*}\), we need to strengthen control measures so as to slow down the TB transmission. In this situation, we take both the treatment of latently infected individuals and infectious individuals, and give the critical value \(m^{*}\) of probability m of the treatment of latently infected individuals so that the basic reproductive number of the model obeys \(R_{0}<1\).
By using the TB data in China, we compare the effect of these different treatment strategies for the control of TB. The numerical simulations totally support our theoretical results. Moreover, our conclusions show that, if the probability γ of successful detection and treatment of infectious individuals cannot reach the critical value \(\gamma^{*}\), only treatment of infectious individuals cannot effectively control the TB transmission, which implies that we should strengthen control measures so as to slow down the TB transmission. That is, we should take both the treatment of latently infected individuals and of infectious individuals.
Declarations
Acknowledgements
This research was supported by National Nature Science Foundation of China Grant 11301314; by Natural Science Basic Research Plan in Shaanxi Province of China Grant 2014JQ1025; by Backbone Youth Training Scheme of Shaanxi University of Science & Technology Grant XSG(4)014; and by National Nature Science Foundation of China Grant 11400856.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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